This proposition comes from that fact that
updating \(\tilde{\pi }(s, a)\)  with respect to a converged \(A(s, a)\) 
leads to \(\tilde{\pi }(a|s)>\pi (a|s)\)  if \( A(s, a)>0,~\forall s, a\)  in actor-critic algorithms [1]}.
Thus, based on Proposition REF , we have the following
\(D_{\mathrm {TV}}^{\mathrm {max}}(\pi , \tilde{\pi }) = \max _{s}\sum _{\begin{array}{c}a\in \mathcal {A}\\ A(s,a)>0\end{array}}[\tilde{\pi }(a|s) - \pi (a|s)]\le \max _{s}\sum _{\begin{array}{c}a\in \mathcal {A} \\A(s,a)>0\end{array}}[M_2 \pi (a|s) - \pi (a|s)]\le M_2 - 1.\) 
